Standard Boolean Forms - Edward Bosworth

Standard Boolean Forms

In this section, we develop the idea of standard forms of Boolean expressions. In part, these forms are based on some standard Boolean simplification rules.

Standard forms are either canonical forms or normal forms.

The standard expressions are in either SOP Sum of Products form, or POS Product of Sums form.

This lecture will focus on the following: Canonical Sum of Products Normal Sum of Products Canonical Product of Sums Normal Product of Sums

We shall also discuss a few more variants that have no standard names.

IMPORTANT: These forms use only the 3 basic Boolean functions: AND, OR, NOT. Specifically, XOR is not used.

Variables and Literals

We start with the idea of a Boolean variable. It is a simple variable that can take one of only two values: 0 (False) or 1 (True). Following standard digital design practice, we use the values 0 and 1. Following standard teaching practice, we denote all Boolean variables by

single letters; normally "A", "B", "C", "D", or "W", "X", "Y", "Z". A literal is either a Boolean variable or its complement.

Literals based on the variable X: X and X . Literals based on the variable Y: Y and Y .

NOTE: X and X represent the same variable, but they are not the same literal. X and Y represent different variables.

Product and Sum Terms

A product term is the logical AND of one or more literals, with no variable represented more than once.

A sum term is the logical OR of one or more literals, with no variable represented more than once.

The following are all valid product terms over the two variables X and Y.

X ?Y

X ?Y

X?Y

X?Y

Forms, such as X?X?Y and X? X ?Y are not considered, as X?X = X and X? X = 0, so X?X?Y = X?Y and X? X ?Y = 0?Y = 0.

The following are all valid sum terms over the two variables X and Y.

X + Y

X + Y

X +Y

X +Y

Single literals According to the strict definition, a single literal is either a sum term or a product term, depending on the context. This is necessary to avoid having to give a number of special cases in the following definitions.

Sum of Products and Product of Sums

A SOP (Sum of Products) expression is the logical OR of product terms. A POS (Product of Sums) expression is the logical AND of sum terms.

Sample SOP expressions

F1(X, Y)

= X?Y + X ?Y

G1(X, Y)

= X ?Y + X?Y

H1(X, Y, Z) = X + Y ?Z

Note: If we did not allow single literals to be product terms, we would have trouble classifying H(X, Y, Z), which is clearly SOP.

Sample POS expressions

F2(X, Y)

= (X+Y) ? ( X +Y )

G2(X, Y)

= ( X +Y) ? (X+Y )

H2(X, Y, Z) = X?(Y + Z)

Note: POS expressions almost always have parentheses to indicate the correct evaluation.

More on Ambiguous Forms

What is the form of the expression F(X, Y) = X + Y 1. SOP It is the logical OR of two product terms.

Each product term is a single literal. 2. POS It is a single sum term (X + Y)

Both statements are true. In general, questions such as this do not concern us. If you are asked a question like this on a test, either answer will be accepted.

This ambiguity comes from the definitional necessity of mentioning "the logical AND of one or more terms" and "the logical OR of one or more terms". With two equally good answers to an ambiguous form, pick the one you like.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download